Hausdorff dimension of univoque sets and Devil's staircase

2017

Nonlinearity

Self Cite

Abstract. To each α ∈ (1/3, 1/2) we associate the Cantor setIn this paper we consider the intersection Γ α ∩ (Γ α + t) for any translation t ∈ R. We pay special attention to those t with a unique {−1, 0, 1} α-expansion, and study the setWe prove that there exists a transcendental number α KL ≈ 0.39433 . . . such that: D α is finite for α ∈ (α KL , 1/2), D αKL is infinitely countable, and D α contains an interval for α ∈ (1/3, α KL ). We also prove that D α equals [0, log 2 − log α ] if and only if α ∈ (1/3, 3− √ 5 2 ]. As a consequence of our investigation we prove some results on the possible values of dim H (Γ α ∩ (Γ α + t)) when Γ α ∩ (Γ α + t) is a self-similar set. We also give examples of t with a continuum of {−1, 0, 1} α-expansions for which we can explicitly calculate dim H (Γ α ∩ (Γ α + t)), and for which Γ α ∩ (Γ α + t) is a self-similar set. We also construct α and t for which Γ α ∩ (Γ α + t) contains only transcendental numbers.Our approach makes use of digit frequency arguments and a lexicographic characterisation of those t with a unique {−1, 0, 1} α-expansion.

Section: Preliminariesmentioning

confidence: 99%

Unique expansions and intersections of Cantor sets

Baker

2017

Nonlinearity

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“…Recently, de Vries and Komornik investigated their topological properties in [8]. Komornik et al considered their Hausdorff dimension in [19], and showed that the dimension function D : q → dim H U q behaves like a Devil's staircase on (1, M +1]. For more information on the univoque set U q we refer to the survey paper [15] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known that U is a Lebesgue null set of full Hausdorff dimension (cf. [6,12,19]). Moreover, the smallest element of U is the Komornik-Loreti constant (cf.…”

Section: Introductionmentioning

confidence: 99%

Univoque bases and Hausdorff dimension

Lü

et al. 2017

Monatsh Math

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“…[11]). Erdős and Joó proved that U has Lebesgue measure zero [14], Daróczy and Kátai [10] later showed that U has full Hausdorff dimension (see also, [18,Theorem 1.6]). Recently, de Vries et al [12,Theorem 1.2] proved that its closure U is a Cantor set, i.e., a nonempty compact set with empty interior and no isolated points.…”

mentioning

confidence: 99%

“…In [18] the authors proved that the entropy function H is a Devil's staircase (see e.g., Figure 1), i.e., H satisfies the following:…”

mentioning

confidence: 99%

Entropy, topological transitivity, and dimensional properties of unique 𝑞-expansions

Barrera

Baker

Trans. Amer. Math. Soc.

2018

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Abstract. Let M be a positive integer and q ∈ (1, M + 1]. We consider expansions of real numbers in base q over the alphabet {0, . . . , M }. In particular, we study the set Uq of real numbers with a unique q-expansion, and the set Uq of corresponding sequences.It was shown in [18, Theorem 1.7] that the function H, which associates to each q ∈ (1, M + 1] the topological entropy of Uq, is a Devil's staircase. In this paper we explicitly determine the plateaus of H, and characterize the bifurcation set E of q's where the function H is not locally constant. Moreover, we show that E is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift (Vq, σ), which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of Uq coincide for all q ∈ (1, M + 1].